Distributional Instruments: Identification and Estimation with Quantile Least Squares

We study instrumental-variable designs where policy reforms strongly shift the distribution of an endogenous variable but only weakly move its mean. We formalize this by introducing distributional relevance: instruments may be purely distributional. Within a triangular model, distributional relevance suffices for nonparametric identification of average structural effects via a control function. We then propose Quantile Least Squares (Q-LS), which aggregates conditional quantiles of X given Z into an optimal mean-square predictor and uses this projection as an instrument in a linear IV estimator. We establish consistency, asymptotic normality, and the validity of standard 2SLS variance formulas, and we discuss regularization across quantiles. Monte Carlo designs show that Q-LS delivers well-centered estimates and near-correct size when mean-based 2SLS suffers from weak instruments. In Health and Retirement Study data, Q-LS exploits Medicare Part D-induced distributional shifts in out-of-pocket risk to sharpen estimates of its effects on depression.

💡 Research Summary

This paper addresses a common problem in instrumental‑variable (IV) analysis: policy reforms often reshape the entire distribution of an endogenous variable (e.g., financial risk) while leaving its mean virtually unchanged. Traditional IV diagnostics focus on mean shifts (the “first‑stage” F‑statistic) and therefore label such designs as weak, even though the instrument may be highly informative about the distribution. The authors formalize this intuition by introducing distributional relevance: an instrument Z is distributionally relevant if Var(E